Download Chapter III: Descriptive Statistics

Chapter VIII:
Elements of
Inferential
Statistics
Hypothesis Testing
• Hypothesis testing is a form of statistical inference to
reach conclusions about scientific problems.
• Hypothesis testing is the mechanism that updates
and tests scientific principles and refines these
principles. This in turn is the basis of the
advancement of scientific research.
• The testing of sample validity is crucial to the
application of inferential statistics, thus we must be
able to test our samples.
Classical/Traditional
Methods
• Classical Hypothesis Testing
o Formal multi-step process that leads to a conclusive statement regarding
the hypothesis:
• Step 1: State Null and Alternate Hypothesis
• Step 2: Select appropriate statistical test
• Step 3: Select level of significance
• Step 4: Delineate regions of rejection and non-rejection of null
hypothesis
• Step 5: Calculate test statistic
• Step 6: Make decision regarding null and alternate hypotheses
State Null and Alternate
Hypothesis
• State null (𝐻0 ) and alternate (𝐻𝐴 ) hypothesis.
• The null is the established value:
o 𝐻0 : 𝜇 = μ𝐻 or 𝐻0 : 𝜇 − μ𝐻 = 0
• This states that the mean value of the null hypothesis is equal to the
actual mean. Thus, if you subtract the hypothesized mean from the
mean then the value should equal zero.
• The alternate hypothesis and the null are mutually
exclusive.
o 𝐻𝐴 comes in three variations, one nondirectional and two directional
• 𝐻𝐴 ≠ 𝐻0 thus the alternate is any value but the null value.
• 𝐻𝐴 < 𝐻0 or 𝐻𝐴 > 𝐻0 thus the alternate is chosen to establish itself as
greater than or less than the null hypothesis depending on the
experiment.
Hypothesis Testing
• Choosing direction
o This graph represents a nondirectional test. The test is only interested if the
alternate hypothesis does not equal the null hypothesis.
o In the case of a < than or > than scenario, the curve would only be
shaded on the side of the alternate hypothesis. Greater than the mean
would shade the right tail, less than the left.
Error in Hypothesis
Testing
• Error in hypothesis testing
o In hypothesis testing we are using a sample to either retain or reject a null
hypothesis. Due to it being from a sample, there is always going to be a
probability of error in rejecting or not rejecting the null hypothesis.
• Type I Error
o A type I error occurs when the null hypothesis is rejected when it is actually
true. The likelihood of this occurring is alpha - 𝛼
o This level is selected prior to the hypothesis test
• Type II Error
o A type II error occurs when the null hypothesis is not rejected but the null
hypothesis is actually false. The likelihood of this error is beta - 𝛽
Selecting a Statistical Test
• An appropriate statistical test must be chosen. In
the chosen case a sample difference of means test
is chosen.
• However, many different tests exist. A list can be found at:
• http://www.graphpad.com/www/Book/Choose.htm
o A sample difference of means test is:
• 𝑍 𝑜𝑟 𝑡 =
•
•
•
•
•
•
𝑥−𝜇
σ𝑥
or
𝑥−𝜇
σ/ 𝑛
Where Z or t is the test statistic ( Z if n > 30, t if n < 30)
𝑥 is the sample mean
𝜇 is the population mean
σ𝑥 is the standard error of the mean
𝜎 is the population standard deviation
n is the sample size
Selecting the Level of
Significance
• The level of significance or alpha (𝛼) can be
established for a hypothesis test.
• The alpha is essentially the probability of error in
your hypothesis test, thus you have control over the
chance of a type I error.
• To reduce the chance of error 𝛼 is often set very low
as errors leading to false conclusions can seriously
flaw research.
• 𝛼 = .05 or .01 are common levels of significance
used.
Selecting Level of
Significance Continued
• Error is very important when setting the 𝛼 of your
hypothesis test.
o The acceptable error of your test must be established based on the needs
of the study.
o Common example in type I errors is a jury trial where an innocent person is
found guilty.
• In this case you would want to establish a very small margin of error if
possible. Is 1 in 20 acceptable? Is 1 in 100?
Delineate Regions of Rejection and
Non-rejection of the Null Hypothesis
• Once 𝛼 has been established we must determine
where we want it to be under the curve.
• Directional or Non-Directional tests are then
revisited to establish where 𝛼 occurs.
• A two-tailed format (nondirectional) must divide 𝛼
between the two tails of the curve thus 𝛼 = .05
represents .025 of each tail region. If 𝛼 = .01 then it
represents .005 in each tail.
• A one-tailed format (directional) establishes the
entire 𝛼 value in one tail depending on whether we
want a < or > value for the alternate hypothesis.
Z-Scores in Hypothesis
Testing
• By converting the 𝛼 into a z-score we can establish
the scores which result in the rejection of the null
hypothesis.
o In a two-tailed test 𝛼 = .025 leaves .95 of the area under the curve.
o Using a table or calculator we can determine the range of .95 to be -1.96
to 1.96 as a z-score.
• Common 𝛼 to z-scores are:
o α = .05 = -1.96, 1.96
o α = .02 = -2.326, 2.326
o α = .01 = -2.576, 2.576
o http://people.richland.edu/james/lecture/m170/ch08-int.html
Calculate the Test Statistic
• Use:
o 𝑍 𝑜𝑟 𝑡 =
𝑥−𝜇
σ𝑥
or
𝑥−𝜇
σ/ 𝑛
o Use the z-score (or t-score) of the test statistic to establish whether the
statistic falls within the selected range of alpha.
o If the score is within the range then the null hypothesis is retained.
o If the score is not within the range then the null hypothesis is rejected.
P-Value Hypothesis
Testing
• Most commonly used approach as classical
approach has limitations.
o In Classical testing:
• Selection of alpha can lack theoretical basis
• Binary rejection/retention of hypothesis is limited
o P-Value Testing:
• Establishes exact probability of getting test statistic essentially the
probability of a type I error
• Establishes a rejection region
P-Value Testing
• Four Steps:
o Calculate Z-Score of test statistic as in classical approach
o Calculate the probability of the value using the area under the normal
curve from a table
o Shade the rejection area by subtracting the probability from value.
o Double the area established if using a nondirectional hypothesis
o A step-by-step example can be found here:
http://www.youtube.com/watch?v=uVvWcFrrvsI
P-Value Testing Continues
• P-Values allow more direct experimental
conclusions:
o Significance and Type I errors are calculated making the exact
measurements available.
o Easy to obtain multiple values over time or area and compare the values
for analysis.
o Still require significant analysis and rigorousness in studies to prevent
invalid results or conclusions.
Applications Using Small
Samples
• Values can be calculated similar to the Z-Test but
using the t-test equations.
o 𝑡=
𝑥−𝜇
𝑠 / 𝑛 −1
or 𝑡 =
𝑥−𝜇
σ/ 𝑛
o Sample sizes less than 30 have a similar distribution with the
exception the there is a higher probability of values falling
in the tails due to the increased uncertainty of the small
sample size in finding the true value.
o Hypothesis rejection is still based on the p-value and if p <
.05 then there is most likely high level of certainty.
One Sample Difference of
Proportions Test
• Z-Tests can be done comparing proportions as well.
o 𝑝 ± 𝑍𝜎𝑝 = 𝑝 ± 𝑧
𝑝(1−𝑝) 𝑁 −𝑛
𝑛 −1
𝑁
o This can establish the proportion to be compared to the population.
o Example of this methodology:
http://www.youtube.com/watch?v=0jeDp03jymQ
Issues in Inferential
Testing and Test Selection
• Defining Degrees of Freedom
o In a sample size of n there are n degrees of freedom
o When a parameter is estimated one degree of freedom is lost.
• In the t-test sample we are estimating 𝜇 and thus one degree of
freedom is lost. This is represented as n - 1 in the equation.
o In two sample tests 𝜇1 and 𝜇2 are calculated so two degrees of freedom
are lost (n – 2).
o Degrees of Freedom is an important concept for many other distributions.
Sampling Issues and
Inferential Testing
• Regardless of sampling method all samples must be
drawn independently and separately.
o Although there are a couple of exceptions not examined here.
• Artificial and Natural Sampling
o Artificial samples draw unbiased random samples from the population
and infers characteristics based on the sample data.
o Natural Samples draw from random events and are analyzed under the
idea that natural events are random processes in themselves. Inferential
processes should be applied with care when applied to natural samples.
o The differences between these methods are important and debated in
Geography.
Inferential Test Selection
• Choosing methods for geographic problems is quite
complicated and requires careful analysis of the
procedural methods.
• Table 8.7 in the text (page 126-127) provides a set to
assist in this process.
• Parametric and NonParametric tests
o Parametric Tests are when inferential tests require knowledge about
population parameters and make assumptions about the underlying
population distribution.
o Non-parametric Tests do not require this underlying knowledge.
Data and Tests
• Parametric tests are run for data at:
o An ordinal or nominal scale; in the case of these scales a parametric test
is required.
o At the intervallic/ratio scale different strategies can be used:
• Run only a parametric test – when no doubt that assumptions and
requirements to run test are met.
• Run only a nonparametric test – when there is reason to believe that
the assumptions or parameters are violated. The data must be
downgraded to a nominal or ordinal scale and then one can run a
parametric test.
• Run both tests – If there is uncertainty in the amount the statistics are
violated. P-tests can be run and the values compared to establish
and investigate the analytical questions.
The End
• Finally, here is an applet designed to demonstrate
classical hypothesis testing for further exploration:
• http://wwwpersonal.umd.umich.edu/~pksmith/JavaStat/Critica
lRegionsPValues.html